Bohr-Favard inequality

An inequality appearing in a problem of H. Bohr [1] on the boundedness over the entire real axis of the integral of an almost-periodic function. The ultimate form of this inequality was given by J. Favard [2]; the latter materially supplemented the studies of Bohr, and studied the arbitrary periodic function

$$ f(x) = \ \sum _ { k=n } ^ \infty (a _ {k} \cos kx + b _ {k} \sin kx) $$

with continuous derivative $ f ^ {(r)} (x) $ for given constants $ r $ and $ n $ which are natural numbers. The accepted form of the Bohr–Favard inequality is

$$ \| f \| _ {C} \leq K \| f ^ {(r)} \| _ {C} , $$

$$ \| f \| _ {C} = \max _ {x \in [0, 2 \pi ] } | f(x) | , $$

with the best constant $ K = K (n, r) $:

$$ K = \sup _ {\| f ^ {(r)} \| _ {C} \leq 1 } \ \| f \| _ {C} . $$

The Bohr–Favard inequality is closely connected with the inequality for the best approximations of a function and its $ r $- th derivative by trigonometric polynomials of an order at most $ n $ and with the notion of Kolmogorov's width in the class of differentiable functions (cf. Width).

References

[1]	H. Bohr, "Un théorème général sur l'intégration d'un polynôme trigonométrique" C.R. Acad. Sci. Paris Sér. I Math. , 200 (1935) pp. 1276–1277
[2]	J. Favard, "Sur les meilleurs procédés d'approximation de certaines classes des fonctions par des polynômes trigonométriques" Bull. Sci. Math. (2) , 61 (1937) pp. 209–224; 243–256
[3]	N.I. [N.I. Akhiezer] Achiezer, "Theory of approximation" , F. Ungar (1956) (Translated from Russian)